A square matrix A is nilpotent if A" = 0 for some positive integer n. == Let V be the vector space of all 2 × 2 matrices with real entries. Let H be the set of all 2 x 2 nilpotent matrices with real entries. Is H a subspace of the vector space V? 1. Does H contain the zero vector of V? choose 2. Is H closed under addition? If it is, enter CLOSED. If it is not, enter two matrices in H whose sum is not in H, using a comma separated list and syntax [12] [56] such as [[1,2], [3,4]], [[5,6], [7,8]] for the answer [³ ¾), § 9]. (Hint: 3 78 to show that H is not closed under addition, it is sufficient to find two nilpotent matrices A and B such that (A+B)" 0 for all positive integers n.) [34] 3. Is H closed under scalar multiplication? If it is, enter CLOSED. If it is not, enter a scalar in R and a matrix in H whose product is not in H, using a comma separated list and syntax such as 2, [[3,4], [5,6]] for the answer 2, 5 6 (Hint: to show that H is not closed under scalar multiplication, it is sufficient to find a real number r and a nilpotent matrix A such that (rA)" #0 for all positive integers n.) 4. Is H a subspace of the vector space V? You should be able to justify your answer by writing a complete, coherent, and detailed proof based on your answers to parts 1-3. choose

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